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In this work, we study two-dimensional Galilean field theories with globaltranslations and anisotropic scaling symmetries. We show that such theorieshave enhanced local symmetries, generated by the infinite dimensional spin-lGalilean algebra with possible central extensions, under the assumption thatthe dilation operator is diagonalizable and has a discrete and non-negativespectrum. We study the Newton-Cartan geometry with anisotropic scaling, onwhich the field theories could be defined in a covariant way. With thewell-defined Newton-Cartan geometry we establish the state-operatorcorrespondence in anisotropic GCFT, determine the two-point functions ofprimary operators, and discuss the modular properties of the torus partitionfunction which allows us to derive Cardy-like formulae.

2d Galilean Field Theories with Anisotropic Scaling